Which expression defines the reflection coefficient Γ for a load ZL on a transmission line with characteristic impedance Z0?

Which expression defines the reflection coefficient Γ for a load ZL on a transmission line with characteristic impedance Z0?

• A. Γ = (ZL−Z0)/(ZL+Z0)
• B. Γ = (Z0−ZL)/(ZL+Z0)
• C. Γ = ZL / Z0
• D. Γ = (ZL+Z0)/(ZL−Z0)

Answer: (A)

The key idea is that the reflection coefficient quantifies how much of the incident voltage wave on a transmission line is reflected when the load does not match the line’s characteristic impedance. At the load, using the boundary conditions for voltage and current, the incident and reflected waves relate to the load impedance ZL and the line impedance Z0 in a way that the ratio of the reflected to incident voltage, Γ = V− / V+, becomes Γ = (ZL − Z0) / (ZL + Z0). This form captures the mismatch: if ZL equals Z0, Γ is zero and there’s no reflection. If ZL is larger than Z0, Γ is positive; if ZL is smaller, Γ is negative. Other forms, like swapping Z0 and ZL or using ZL/Z0 or (ZL+Z0)/(ZL−Z0), do not satisfy the boundary conditions and misrepresent the relation between the load and line impedances. Hence the correct expression is Γ = (ZL − Z0) / (ZL + Z0).

The key idea is that the reflection coefficient quantifies how much of the incident voltage wave on a transmission line is reflected when the load does not match the line’s characteristic impedance. At the load, using the boundary conditions for voltage and current, the incident and reflected waves relate to the load impedance ZL and the line impedance Z0 in a way that the ratio of the reflected to incident voltage, Γ = V− / V+, becomes Γ = (ZL − Z0) / (ZL + Z0). This form captures the mismatch: if ZL equals Z0, Γ is zero and there’s no reflection. If ZL is larger than Z0, Γ is positive; if ZL is smaller, Γ is negative. Other forms, like swapping Z0 and ZL or using ZL/Z0 or (ZL+Z0)/(ZL−Z0), do not satisfy the boundary conditions and misrepresent the relation between the load and line impedances. Hence the correct expression is Γ = (ZL − Z0) / (ZL + Z0).